CRR Framework: Moss Biology (Bryum capillare)

d/dt(∂L/∂ẋ) - ∂L/∂x = ∫₀ᵗ K(t-τ)φ(x,τ)e^(C(x,τ)/Ω)Θ(t-τ) dτ + Σᵢ ρᵢ(x)δ(t-tᵢ)

Complete CRR Mathematical Implementation & Code Mapping

DIRECT MATHEMATICAL IMPLEMENTATION:

This simulation explicitly computes:
• Lagrangian density L(x,t) from biomass, carbon, reproduction, stress memory
• Coherence C(x,t) = ∫₀ᵗ L(x,τ)dτ via temporal integration
• Memory kernel K(t-τ) = λe^(-λ(t-τ)) via exponential relaxation
• Rebirth weight exp(C/Ω) directly in growth calculations
• Rupture conditions as threshold crossings → δ(t-tᵢ) events

1. LAGRANGIAN DENSITY L(x,t)

Mathematical Definition:

                                L(x,t) = L_biomass + L_carbon + L_reproduction + L_memory

                                where:

                                L_biomass = (protonema + gametophore + buds + rhizoids) × w_biomass

                                L_carbon = carbon_density × w_carbon

                                L_memory = growthMemory × w_mem

Code Implementation:

computeLagrangian(idx) {

  const biomass = this.protonema[idx] + this.gametophore[idx] + 

                this.buds[idx] + this.rhizoids[idx];

  const carbon = this.carbon[idx];

  const memory = this.growthMem[idx];

  return biomass * 1.0 + carbon * 0.8 + memory * 0.3;

}

Physical Interpretation: The Lagrangian measures "organized biological activity" - higher L indicates more structured, energy-rich, reproductively capable moss tissue.

2. COHERENCE FUNCTIONAL C(x,t) = ∫₀ᵗ L(x,τ)dτ

Mathematical Definition:

                                C(x,t) = ∫₀ᵗ L(x,τ) dτ
                            

Accumulated "action" or "organized complexity" over system history

Discrete Time Integration:

                                C(x, t+Δt) = C(x,t) + L(x,t) × Δt
                            

Code Implementation:

updateCoherence(dt) {

  for (let i = 0; i < this.cells; i++) {

    this.lagrangian[i] = this.computeLagrangian(i);

    this.coherence[i] += this.lagrangian[i] * dt;

    if (this.coherence[i] > 10) {

      this.coherence[i] = 10 + Math.log(1 + this.coherence[i] - 10);

    }

  }

}

Key Property: C is monotonically increasing between ruptures (dC/dt = L ≥ 0), and only resets/jumps at rupture events. This accumulated history exponentially weights future dynamics.

3. MEMORY KERNEL K(t-τ) = λe^(-λ(t-τ))

Mathematical Form:

                                K(t-τ) = λ exp(-λ(t-τ))
                            

Exponential memory decay with characteristic time 1/λ

Integral Form (what appears in CRR equation):

                                ∫₀ᵗ K(t-τ) × signal(τ) dτ
                            

Equivalent Differential Form:

                                dmemory/dt = λ(signal - memory)
                            

This first-order ODE has the same solution as the integral form

Code Implementation:

updateMemory(lambda) {

  for (let i = 0; i < this.cells; i++) {

    const totalBiomass = this.protonema[i] + this.gametophore[i] + 

                        this.buds[i] + this.rhizoids[i];

    this.moistureMem[i] += lambda * (this.moisture[i] - this.moistureMem[i]);

    this.nutrientMem[i] += lambda * (this.nutrients[i] - this.nutrientMem[i]);

    this.growthMem[i] += lambda * (totalBiomass - this.growthMem[i]);

  }

}

Default λ = 0.025: This gives memory timescale τ_mem = 1/λ = 40 timesteps. Past signals decay to ~37% after 40 steps, ~13.5% after 80 steps.

4. REBIRTH WEIGHTING exp(C/Ω)

Mathematical Definition:

                                R_χ(x,t) = ∫₀ᵗ φ(x,τ) exp(C(x,τ)/Ω) Θ(t-τ) dτ
                            

Regeneration/growth weighted exponentially by accumulated coherence

Code Implementation:

// Growth rate calculation (simplified example):

const coherenceWeight = Math.exp(fields.coherence[idx] / this.omega);

const baseGrowthRate = nutrientSignal * moistureSignal * (1 - stress);

const effectiveGrowthRate = baseGrowthRate * coherenceWeight;

Physical Meaning: Regions with high accumulated coherence (C) regenerate/grow exponentially faster. This creates positive feedback: successful regions become more successful, leading to self-organized patterns.

⚠️ Numerical Stability: Since exp(C/Ω) can grow unbounded, we cap coherence or use soft saturation: exp(C/Ω) → tanh(C/Ω) for large C, or enforce rupture before explosion.

5. RUPTURE δ(t-tᵢ) IMPLEMENTATION

Mathematical Definition:

                                Σᵢ ρᵢ(x) δ(t-tᵢ)
                            

Discrete discontinuous jumps at rupture times {tᵢ} with amplitudes ρᵢ(x)

Rupture Condition:

                                Rupture occurs when: stress(t) > θ_effective(x,t)

                                where: θ_effective = θ_base × (1 + μ × stressMemory)

Code Implementation:

// Check rupture condition:

const stress = this.sim.env.getStress();

const adaptiveThreshold = STRESS.RUPTURE * (1 + cell.stressMemory * 0.3);

if (stress.combined > adaptiveThreshold && Math.random() < ruptureProb) {

  // RUPTURE EVENT

  cell.energy *= 0.7; // amplitude ρᵢ = -0.3 × energy

  fields.coherence[idx] *= 0.5; // partial coherence loss

  // This is the δ(t-tᵢ) discontinuity

}

Biological Mapping: Cell death (energy → 0), tissue damage (biomass reduction), reproductive dispersal (spore release) are all rupture events with different ρᵢ amplitudes.

COMPLETE IMPLEMENTATION MAPPING:

CRR Mathematical Term	Code Implementation	Status
L(x,t)	computeLagrangian() from biomass + carbon + memory	✓ Direct
C(x,t) = ∫₀ᵗ L dτ	coherence[i] += L(i) × dt each timestep	✓ Direct
K(t-τ) = λe^(-λ(t-τ))	memory += λ(signal - memory)	✓ Exact (differential form)
exp(C/Ω)	Math.exp(coherence / omega)	✓ Direct
ρᵢ(x)δ(t-tᵢ)	Threshold crossing → discrete state jump	✓ Discrete approx
φ(x,τ)	moisture × nutrients × (1-stress) × (1-density)	✓ Phenomenological

Key Point: This is not an analogy or metaphor. The CRR mathematical objects (L, C, K, exp(C/Ω), δ) are explicitly computed in the simulation code. The moss behavior emerges from these CRR dynamics.

CRR Framework: Moss Biology (Bryum capillare)

Visualisation Mode

Environmental Parameters

Colony Metrics

CRR Metrics (Computed)

Environmental Health

CRR Parameters